1. Field of the Invention
The invention relates to a method of reconstructing raw data to produce magnetic resonance image data of an object under examination, as well as a method of producing magnetic resonance image data of an object under examination based on raw data acquired by a magnetic resonance system, using such a reconstruction method. Furthermore, the invention relates to a reconstruction device for reconstructing raw data to produce magnetic resonance image data of an object under examination, as well as a magnetic resonance system having such a reconstruction device.
2. Description of the Prior Art
In a magnetic resonance system, the subject to be examined is exposed by a base field magnet system to a relatively high basic field magnetic field, for example, of 1.5 Tesla, 3 Tesla or 7 Tesla. After applying the basic field, nuclei in the object under examination align along the field with a non-vanishing nuclear magnetic dipole moment, frequently also called spin. This collective performance of the spin system is described as macroscopic “magnetization”. Macroscopic magnetization is the vector sum of all microscopic magnetic moments at a specific location in the object. In addition to the basic field, a gradient system, superimposes a magnetic field gradient thereon, allowing the magnetic resonance frequency (Larmor frequency) of nuclear spins at a respective location to be determined. By appropriate antenna devices, radio frequency excitation signals (RF pulses) are emitted via a radio frequency transmission system, which result in the nuclear spins of specific nuclei being resonantly excited (i.e., at the Larmor frequency that exists at the specific location) so as to be deflected by a defined flip angle with respect to the magnetic field lines of the basic magnetic field.
If such an RF pulse has an effect on spins that are already excited, they can be flipped to a different angular position, or even forced back to an original position parallel to the basic magnetic field. In the process of relaxing the excited nuclear spins, radio frequency signals (magnetic resonance signals) are resonantly emitted, which are received by an appropriate receiving antenna (also called magnetic resonance coils). The received signals are subsequently demodulated and digitized and processed as “raw data”. The acquisition of magnetic resonance signals is performed in the spatial frequency domain, called “k-space”. For this purpose, the acquired raw data are entered into a memory organized as k-space, at respective sample points in k-space along a trajectory or path, comprised of sample points, that is defined by the combination of the gradients that are activated during readout. In addition, the RF pulses have to be emitted in temporally appropriate coordination. After further procedural steps, which usually depend also on the acquisition method, the desired image data can be reconstructed from the raw data thus acquired, by a two-dimensional Fourier transformation. It is also possible to excite and read, in a defined manner, three-dimensional volumes, wherein after further procedural steps the raw data are entered into a memory organized as three-dimensional k-space. Correspondingly, it is possible to reconstruct a three-dimensional image data volume by a three-dimensional Fourier transformation.
Typically, to control a magnetic resonance imaging system, specific predetermined pulse sequences, i.e., successions of defined RF pulses, as well as gradient pulses in different directions and readout windows, are used for measurements, while the receiving antenna are set for reception and the magnetic resonance data are received and processed. By a measurement protocol, these sequences are parameterized in advance for a desired examination, for example, to obtain a specific contrast of calculated images. In addition, the measurement protocol may include further control data for the measurement. A large number of magnetic resonance sequence techniques exist, based on which pulse sequences can be constructed. One of the great challenges of the future development of magnetic resonance imaging (MR imaging) is to accelerate the data acquisition in such magnetic resonance sequence techniques without significantly compromising resolution, contrast and artifact susceptibility.
Current clinical MR imaging is almost exclusively based on Cartesian or rectilinear imaging, in which the scanned k-space points (i.e., the sample points in the k-space where raw data are entered) are located on the grid points of a rectangular grid or screen. It is possible to accelerate significantly such clinical imaging using parallel imaging methods. In parallel MR imaging, data acquisition is reduced by not acquiring a portion of the lines of the grid that are actually required for reconstructing a non-convolving image in k-space. These missing lines are later substituted during image reconstruction in k-space, or the artifacts that result from such undersampling are removed from k-space. To use the parallel imaging methods, it is required to receive the radio frequency signals with multiple receiving coils (antenna), and the spatial sensitivity of the individual receiving coils has to be known. The spatial sensitivity of the individual receiving coils is calculated from coil calibration data. Usually, the coil calibration data has to be sufficiently scanned. Because of the fact that the spatial sensitivities vary gradually, the coil calibration data generally requires a low spatial resolution. In general, the coil calibration data has to be re-measured for each patient. One of the most important parallel imaging methods is the GRAPPA method, which is described, for example, in the article “Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA)” by Marc Griswold et al., published in Magnetic Resonance in Medicine 47, 2002, pp. 1202 to 1210. The “missing” raw data si (ky,kx) of the coil i at the k-space position k=(ky,kx) are calculated or interpolated with the k-space coordinates (ky,kx) for which no data was acquired as a linear combination of all measured data points in a specified vicinity or neighborhood Ω(ky,kz) of the missing sample points:
                                          s            i                    ⁡                      (                                          k                y                            ,                              k                x                                      )                          =                              ∑                          j              =              1                                      N              c                                ⁢                                    ∑                                                (                                                            q                      y                                        ,                                          q                      x                                                        )                                ∈                                  Ω                                      (                                          ky                      ,                      kx                                        )                                                                        ⁢                                                            n                                      i                    ,                                          (                                                                        k                          y                                                ,                                                  k                          x                                                                    )                                                                      ⁡                                  (                                      j                    ,                                          q                      y                                        ,                                          q                      x                                                        )                                            ⁢                                                s                  j                                ⁡                                  (                                                            q                      y                                        ,                                          q                      x                                                        )                                                                                        (        1        )            wherein i and j and the indices for the individual receiving coils used in parallel measurements, and respectively extend from 1 to Nc, the maximum of receiving coils used. The external (first) sum in the formula (1) integrates all receiving coils, the internal (second) sum integrates all “measured” sample points on which raw data was acquired and which are located in a definite neighborhood Ω(ky,kz) of the respectively “missing” sample points with the k-space coordinates (ky,kx). sj(qy,qx) is the respective signal (i.e., the raw data acquired there) measured by the j-th receiving coil at the sample point with the k-space coordinates (qy, qx). ni,(ky,kx) are the complex linear factors which assess the individual data points measured in the vicinity Ω(ky,kz), and which are initially unknown. The index {i,(ky,kz)} indicates that in general a separate set of linear factors is required not only for each coil i but also for each unmeasured data point with the coordinates (ky,kx).
The basis of this method is that the coefficients or weighting factors ni,(ky,kx) (subsequently also called “GRAPPA weights”) in the formula (1) for rectangular imaging are independent of the location (ky,kx) of the sample point on the grid, and depend only on the distances to the respective neighboring sample points to be considered:
                                          s            i                    ⁡                      (                                          k                y                            ,                              k                x                                      )                          =                              ∑                          j              =              1                                      N              c                                ⁢                                    ∑                              l                =                0                                                              N                  y                                -                1                                      ⁢                                          ∑                                  m                  =                  0                                                  Nx                  -                  1                                            ⁢                                                                    n                    i                                    ⁡                                      (                                          j                      ,                      l                      ,                      m                                        )                                                  ⁢                                                      s                    j                                    ⁡                                      (                                                                                            k                          y                                                +                                                                              (                                                                                          A                                ⁢                                                                                                                                  ⁢                                1                                                            -                                                              1                                0                                                                                      )                                                    ⁢                          Δ                          ⁢                                                                                                          ⁢                                                      k                            y                                                                                              ,                                                                        k                          x                                                +                                                                              (                                                          m                              -                                                                                                N                                  x                                                                2                                                                                      )                                                    ⁢                          Δ                          ⁢                                                                                                          ⁢                                                      k                            x                                                                                                                )                                                                                                          (        3        )            wherein Δky is the grid spacing (the grid dimensions) between adjacent sample points in the phase coding direction, Δkx is the grid spacing between adjacent sample points in the frequency coding direction, and A is the acceleration factor, l and m are control variables of the neighboring sample points. l0 is selected so that all sample points were measured on the right side of equation (3) and involve neighboring sample points of si. ni, in turn, involve the complex linear factors, which estimate the respectively measured data points in the vicinity and which are initially unknown. In equation (3), the rectangular vicinity of each unmeasured data point Nx×Ny is composed of data points that were individually acquired (detected) with Nc different component coils. Because of the fact that on the left side of equation (3), the unmeasured data for each component coil are calculated separately and the linear factors for various component coils are different, a total of Nunknown=Nc·Ny·Nx·NC complex GRAPPA weights are required for reconstructing the unmeasured data. The GRAPPA weights are obtained by measuring a second data set, called the “coil calibration data set”. This coil calibration data set is completely (i.e., adequately according to the Nyquist theorem) scanned or measured. Because of the complete scanning process, the raw data si(ky,kx) on the left side of equation (3) as well as the raw data sj(qy,qx) on the right side of equation (3) are known. If the coil calibration data set has at least as many data points as unknown GRAPPA weights are available, it is possible to calculate the GRAPPA weights. For this purpose, it is easiest to write equation (3) for each component coils in matrix format:si=G·ni  (2)
 In this format, ni is a column vector of the length Ny·Nx·NC the components of which include the required GRAPPA weights for coil i. The column vector si is a vector consisting of M data points of the coil calibration data set for which all neighbors in the selected vicinity have also been measured. Consequently, the column vector si has the length M and includes only data points of the selected component coil i. As a result, G involves an M×Ny·Ny·NC matrix. The elements of the matrix G consist of measured data points. Consequently, the m line of matrix G consists of the total Ny·Nx·NC data points in the rectangular vicinity of the m data point according to equation (3).
Usually, it is necessary to measure so many sample points that the equation system is overdetermined. Subsequently, this equation system is resolved with standard methods in terms of the smallest squared deviation.
In addition to Cartesian imaging, radial imaging is gaining increasing significance, primarily because of its relative insensitivity to movement of the subject. In radial imaging, data acquisition is performed along radial spokes that proceed through the k-space center. The relative insensitivity to movement is because of the repeated acquisition of central k-space. A primary disadvantage of radial imaging is that because of the oversampling of the central k-space region, the required data volume for reconstructing an artifact-free image is higher by at least a factor π/2. Therefore, acceleration of the data acquisition is a special requirement for achieving a broad clinical acceptance of this radial technique.
In principal, it is possible to use parallel imaging techniques, like the above-mentioned GRAPPA, even in radial imaging, to reduce the acquisition periods. However, in non-Cartesian acquisition, usually a separate set of GRAPPA weight is required for each “missing” sample point. Provided that for each of these missing sample points, an appropriate densely measured coil calibration data set is available, the numerical quantity increases linearly with the number of unmeasured sample points which, in turn, is proportional to the total number of sample points on the grid.
From the abstract “Direct Parallel Imaging Reconstruction of Radially Sampled Data Using GRAPPA with Relative Shifts” by Mark Griswold et al., published in Proc. Intl. Soc. Mag. Reson. Med. 11 (2003) with the program number 2049, a method is known that reduces the numerical quantity compared to an exact GRAPPA reconstruction for a radial acquisition diagram. However, it is assumed that a radial grid within a concentric ring can be exchanged approximately with a Cartesian grid, making it possible to use a set of GRAPPA weights for all missing data points with the same or a similar radial distance from the k-space center. This simplified assumption can result in an incomplete development of, or increase in noise in, the reconstructed images, so such images cannot always be suitable for practical use.